Question

The dimensions of the quantities in one (or more) of the following pairs are the same. Identify the pair(s).
(Given, $\text{Torque}=\text{Force}\times \text{distance}$,
$\text{Angular momentum}=\text{momentum}\times \text{distance}$
$\text{Young's modulus}=\frac{stress}{strain},\text{Stress}=\frac{\text{Force}}{\text{Area}},\text{Strain}=\frac{\text{Change in length}}{\text{Original length}}$

• A. Torque and work
• B. Angular momentum and work
• C. Energy and Young’s modulus
• D. Light year and wavelength

Answer 1

Answer: (A), (D)

Light year and wavelength

For torque and work:
Torque$\left(T\right)=Force\times distance$ ........ (given)
We know, Work done $\left(W\right)=Force\times distance$
So dimensions of torque and work must be same.

For angular momentum and work:
$W=Force\times distance=\left[M{L}^2{T}^{-2}\right]$
Angular momentum $=momentum\times distance=\left[ML{T}^{-1}\right]\left[L\right]=\left[M{L}^2{T}^{-1}\right]$

For energy and Young's modulus:
$\left[E\right]=\left[M{L}^2{T}^{-2}\right]$
for Young's modulus, $\left[Y\right]=\left[stress\right]$, as strain is dimensionless
$\left[Y\right]=\left[\frac{force}{area}\right]=\frac{\left[ML{T}^{-2}\right]}{\left[L^2\right]}=\left[M{L}^{-1}{T}^{-2}\right]$

For light year and wavelength:
Both quantities are measured over distance, hence there dimensions are equal.